Uncertainty and Disagreement in Economic Forecasting∗

Stefania D’Amico and Athanasios OrphanidesBoard of Governors of the Federal Reserve System

March 2006

∗Preliminary and incomplete draft, prepared for the conference in honor of Charles R.Nelson, Federal Reserve Bank of Atlanta, Atlanta, Georgia, March 31-April 1, 2006. Theopinions expressed are those of the authors and do not necessarily reflect the views of theBoard of Governors of the Federal Reserve System.

Correspondence: D’Amico: Federal Reserve Board, Washington, D.C. 20551, Tel.: (202)452-2567, e-mail: Stefania.D’Amico@frb.gov. Orphanides: Federal Reserve Board, Wash-ington, D.C. 20551, Tel.: (202) 452-2654, e-mail: Athanasios.Orphanides@frb.gov.

1 Introduction

Individual decisions, aggregate economic outcomes, and asset prices are often shaped by

probabilistic assessments of future events, the uncertainty surrounding them, and disagree-

ment across individuals regarding these assessments. The roles of disagreement and un-

certainty are often confounded in economic analysis, however, in large part because the

observation and measurement that would be needed for such investigations is lacking. For

instance, it is unclear from the available evidence whether variations over time in the risk

and term premia embedded in bond and equity prices primarily reflect changes in aggregate

uncertainty or swings in the degree of the consensus in beliefs about economic fundamentals.

Surveys of expectations offer one potential source for information that could be useful

in addressing such questions. Unfortunately, virtually all surveys of analysts, professional

forecasters, businesses and households collect information on point predictions of future

events. Aggregating information on the most likely outcome in each individual’s assessment

offers some indication of the degree of consensus among individuals regarding the most likely

outcome. It does not, however, provide meaningful information regarding the uncertainty

that each individual may attach to his or her point forecast or the evolution of the associated

individual and aggregate uncertainty over time, or any disagreements regarding individual

assessments of uncertainty.

The Survey of Professional Forecasters presents a useful exception to the typical sur-

vey structure. Since its inception, this quarterly survey has asked respondents to provide

probabilistic assessments of the outlook for inflation and for either nominal or real output.

Focusing on these probabilistic responses allows construction and limited comparisons of

aggregate proxies for uncertainty and disagreement. Starting with the important study by

Zarnowitz and Lambros (1988) a number of authors have suggested various approaches to

measure uncertainty as reflected in these survey responses.1 In this paper, we extend earlier

work and present quarterly time series of various measures of uncertainty and disagreement

that correct some apparent biases due to imperfections in the survey.

We focus our attention on the probabilistic inflation forecasts that cover an almost con-1Recent studies examining these density forecasts and their properties include Rich and Tracy (2003),

Giordani and Soderlind (2003). Lahiri and Liu (2004), Wallis (2005), Engelberg, Manski and Williams(2006).

1

tinuous span approaching forty years, from 1968Q4 to 2006Q1. Following earlier work, our

starting point is the estimation of the characteristics of the probability densities for each

forecaster in each period available in the survey. An important difficulty, however, is that

the design of the survey and individual respondent practices, particularly a tendency to ap-

proximate and round the probabilistic responses, induces some errors in the estimation of

individual uncertainty. These errors appear especially problematic when individual uncer-

tainty is small and can result in systematic biases in estimates of uncertainty. To arrive at

an aggregate characterization of uncertainty that mitigates this problem, we propose robust

estimation of a cumulative density function approximating the distribution characterizing

individual uncertainty in each quarter. Doing so provides direct estimates of both the aver-

age degree of uncertainty as well as the dispersion of individual uncertainty assessments in

each quarter. This allows comparing and contrasting average uncertainty not only with dis-

agreement about the mean outlook, (the most commonly discussed form of disagreement),

but also with disagreement about the uncertainty to the outlook.

As an illustration of the potential usefulness of distinguishing uncertainty from various

forms of disagreement we also examine their relationship with and predictive content for the

evolution of the term premia embedded in the term-structure of government securities. We

show that all three measures of uncertainty and disagreement we discuss are significantly

correlated with term premia. However, their comovement in the sample makes it difficult

to identify their relative importance with precision.

2 Description of the Survey Data

The first Survey of Professional Forecasters was conducted in November 1968 as a joint effort

by the American Statistical Association and the National Bureau of Economic Research and

quarterly surveys have been conducted since then. Currently, the Survey is maintained by

the Federal Reserve Bank of Philadelphia that took over its administration in the summer of

1990.2 The number of respondents has changed over time and, especially in its early years,

fluctuated considerably. As Figure 1 indicates, however, with the exception of the late2See Zarnowitz and Braun (1987) and Croushore (1993) for useful historical descriptions and assessments

of the survey.

2

1980s, the survey has included between 20 and 40 regular respondents, that is respondents

who have contributed to it multiple times over a period of one or more years.

In the following, we highlight some special features relating primarily to the density

forecasts that are of importance for our analysis. Since the inception of the survey, re-

spondents have been asked to provide probabilistic forecasts for inflation, as measured by

the concept of aggregate output most relevant at the time of each survey (that is GNP

or GDP with fixed or chain weights). A complicating factor in the design of the survey

is that in each quarter respondents are asked to forecast the annual percentage changes

in the average output deflator (currently the GDP price index) between the previous and

current year between the current and following year. When the survey started, there was

one question referring to of only one of these annual forecasts, but since the early 1980s,

forecasts for both have generally been asked. Because the questions aways to the average

of the calendar year, their structure introduces a seasonal pattern in the quarterly horizons

relevant for the forecast. For example, in the first quarter of the year, the current year

forecast is a 4-quarter ahead forecast, counting from the last quarter of available data (the

fourth quarter of the previous year) to the last quarter of the current year. The following

year forecast reflects an 8-quarter ahead horizon, ending with the fourth quarter of the

subsequent year. By contrast, in the second quarter of the year, these horizons become

3 and 7 quarters ahead, respectively. Similarly, in the third quarter the forecast horizons

are 2 and 5 quarters ahead, and finally, in the fourth quarter of the year, they are down

to 1 and 4 quarters ahead, respectively. As a result of this design in the survey the un-

derlying expected uncertainty reflected in the individual responses as well as the expected

degree of disagreement have seasonal patterns. An approximate year-ahead horizon can

be constructed by using a time series that concentrates on the 4-quarter ahead horizon in

first quarters, the 3-quarter ahead horizon in second quarters, the 6-quarter ahead horizons

in third quarters and the 5-quarter ahead in fourth quarters. Nonetheless, an element of

seasonality would certainly remain in this quarterly series.

The presence of this seasonality, in conjunction with the way the questions are posed

and responses presented introduces some additional complications. In each quarter, the

survey presents respondents with a number of bins denoting spanning a wide range of

3

outcomes for the annual rate of inflation and asks them to place probabilities in those bins,

adding to 100%. In recent years, the survey has included 10 bins, with width equal to

one percent, except for the edges that are open ended. Figure 2 shows two examples of

individual responses from this period in the form of the histogram. The example on the left

shows a respondent who placed probability mass in all available bins. This would be the

expected outcome if the underlying distribution were continuous with wide support and the

responses were provided in rather exact terms, say in multiples of 1 percent or fractions of

1 percent. This, however, is not the typical practice. Most of the time, most respondents

place all their density in a few bins only. Some pertinent summary statistics for all responses

since 1992 are presented in Figures 3–5. As can be seen in Figure 3, the distribution of the

number of non-zero-bins depends sensitively on the forecast horizon. At one extreme, when

the horizon is just one quarter, almost 60 percent of respondents place all probability mass

in just one or two bins. At the other extreme, when the horizon is 8 quarters, about 60

percent of respondents use 4 or more bins in their answers.

An important reason for this pattern in the number bins employed is likely the manner

in which respondents may be rounding the answers they provide. Examination of the

individual entries reveals that quite often the probabilities in all non-zero bins are multiples

of 10 percent or multiples of 5 percent. The breakdown of this practice, by horizon, is shown

in Figure 4. The relationship of rounding with the number of bins employed is reported in

Figure 5. In each case, responses are classified as reflecting 10% rounding if all non-zero bins

are multiples of 10, 5% rounding if all non-zero bins are multiples of 5 and 1% rounding if

they do not fall in either of the other two categories. As can be seen in Figure 4, almost 80

percent of two-bin responses are multiples of 10 percent, like the example on the right panel

of Figure 2. By contrast, only half of responses that spread to 5 or more bins are reported in

multiples of either 5 or 10 percent. This pattern suggests that answers are often provided in

approximate form but also, since this approximations are more coarse when fewer bins are

utilized, that it may disproportionately influence estimates of individual uncertainty when

it is relatively low.

Some other issues also generate difficulties that need to be addressed in constructing

quarterly estimates of disagreement and uncertainty that have a consistent interpretation

4

over time. One is the changing number and width of the intervals over time. Fewer and

larger bins induce greater biases. The survey initially offered 15 bins with each interval

having a width of 1%. But from 1981 to 1991 only 6 bins with a width of 2% were presented

to respondentd. Since 1992, the probabilistic questions offer 10 bins whose width equal to

1%.

Another complication relates to changes in the number and composition of the respon-

dents, which was shown in Figure 1. In early surveys the average number of respondents

was around 60, while in recent years the average has been closer to 30. In addition, at times

the turnover of the SPF forecasters has been quite high. To eliminate possible biases from

these variations, attention could be restricted to a subset of “regular” forecasters who con-

tributed consistently over a certain number of years. Zarnowitz and Lambros, for example,

restricted attention to forecasters who participated in at least 12 surveys. To check the

robustness of our results against such participation criteria, we computed and compared

our measures using samples including only respondents who participated in at least 4, 8

and 12 surveys, in addition to the sample of all respondents.

Finally, errors in the survey introduce missing observations in some quarters. For ex-

ample, there is some uncertainty about what years respondents were providing probabilistic

responses in 1985:Q1 and 1986:Q1 so these two quarters are excluded from the analysis.

3 Parameter-Free Analysis of the Probabilistic Beliefs

In this section we present some direct measures of uncertainty and disagreement that do

not assume any specific continuous distributions for the probabilistic beliefs but are simply

based on different ways of computing sample means and variances. The first step needed

to compute these measures is to make some assumptions about the range over which the

individual histograms are defined. We assume that the first and last intervals (that are open

ended in the survey) are closed and have the same width as that of the central intervals.

Results are generally not be very sensitive to this assumption since most of the times

respondents place zero probability in the extreme intervals. When they do place some

probability at the edges, a potential bias is introduced. The second step requires making a

choice regarding the concentration of the probability mass within each interval. We adopt

5

two common alternatives, either that the frequencies are concentrated at the midpoints

of the intervals, or that the probability mass is uniformly distributed within the interval

(Zarnowitz and Lambros (1987)). Both alternatives entail a degree of approximation that

also depends on the width of the intervals and bias upwards estimates of the individual

variances. To compensate for this, we apply the Sheppard correction to variance estimates

from these two methods (see Kendall and Stuart (1977)), although we recognize that this

correction is much less accurate for the uniform assumption relative to the alternative that

places all density within an interval at the midpoint of the interval.

To compute the mean (µ) and variance (v) of the individual histograms for the mid-point

(M) and the uniform (U) assumption we apply the following formulas, respectively:

µMi,h,t =

n∑

j=1

xj,h,t · pi,j,h,t,

vMi,h,t =

n∑

j=1

(xj,h,t − µM

i,h,t

)2 · pi,j,h,t

− w2

t

12,

µUi,h,t =

n∑

j=1

(uj,t − lj,t)2

· pi,j,h,t,

vUi,h,t =

n∑

j=1

(u3

j,t − l3j,t

)

3 (uj,t − lj,t)· pi,j,h,t −

(µU

i,h,t

)2

− w2

t

12,

where pi,j,h,t is the probability that the forecaster i assigns to the jth interval in the survey

conducted at time t with horizon h; uj,t and lj,t are the upper and lower limits of the jth

interval; and wt is the width of the central range, thus the entire term w2t

12 represents the

Sheppard’s correction for the second moment.

To obtain aggregate measures of the mean expectation of inflation and uncertainty about

inflation for a specific quarter, the individual means and variances are averaged across all

Nt respondents:

µMh,t =

1Nt

Nt∑

i=1

µMi,h,t,

vMh,t =

1Nt

Nt∑

i=1

vMi,h,t,

6

µUh,t =

1Nt

Nt∑

i=1

µUi,h,t,

and

vUh,t =

1Nt

Nt∑

i=1

vUi,h,t.

In order to compute the measure of disagreement, we calculate the cross-sectional stan-

dard deviation of the individual means obtained using both assumptions:

sMh,t =

√√√√ 1Nt

Nt∑

i=1

(µM

i,h,t − µMh,t

)2,

and

sUh,t =

√√√√ 1Nt

Nt∑

i=1

(µU

i,h,t − µUh,t

)2.

Having computed the variances of the individual histograms, it is also possible to quan-

tify a further concept of disagreement, that is the dispersion of the second moments of the

probabilistic beliefs rather than the fist moments. This measure, which can be interpreted

as disagreement about inflation uncertainty, can be computed as follows:

φMh,t =

√√√√ 1Nt

Nt∑

i=1

(vMi,h,t − vM

h,t

)2,

and

φUh,t =

√√√√ 1Nt

Nt∑

i=1

(vUi,h,t − vU

h,t

)2.

4 Parametric analysis of the probabilistic beliefs

Next we examine parametric methods for the analysis of the probabilistic beliefs. This

can be divided in two parts. In the first part of the analysis we examine assumptions on

the distribution of inflation faced by an individual, while in the second part we examine

assumptions about the distribution of inflation uncertainty across individuals.

For the first part, we focus on the assumption that a normal distribution approximates

the probabilistic beliefs of each respondent sufficiently well (Giordani and Soderlind (2003)).

7

As we discuss below, however, this assumption provides useful estimates only for forecasters

that assign positive probability to three or more bins. Initially, we also experimented with

fitting alternative distributions that accommodated deviations from normality. However,

we concluded that it was difficult to distinguish between true deviations from normality

and apparent deviations due to the approximation and rounding of individual responses.

Consequently, we decided to focus on the first two moments of the individual probabilistic

distributions, and limited attention to the normal density.

For each individual probabilistic response, we fit the Normal CDF so that it matches

as closely as possible the individual empirical CDF obtained by accumulating the densities

in individual bins. More precisely, we identify the mean and variance of the Normal that

minimize the sum of the squared differences between the empirical CDF and the probabilities

implied by the normal distribution over the same intervals. The resulting estimates of the

mean and the standard deviation of the normal (N) constitute the estimate of the expected

inflation and of the individual uncertainty respectively, that is if

Xi,h,t ∼ N(µi,h,t,;σ2

i,h,t

),

then

µNi,h,t = µi,h,t; vN

i,h,t = σ2i,h,t,

Assuming that this procedure produces meaningful estimates for all individuals, we can

aggregate the resulting individual results as in the previous section:

µNh,t =

1Nt

Nt∑

i=1

µNi,h,t; vN

h,t =1Nt

Nt∑

i=1

vNi,h,t.

However, fitting normal densities to individual histograms provides meaningful estimates

of the underlying distribution only when when forecasters assign positive probabilities to

at least 3 bins. To illustrate the difficulty encountered with fewer bins, in is instructive to

compare the estimation associated with the two example empirical densities shown in Figure

6. On the left, a unique best fitting normal distribution can be easily identified. On the right,

however, the two bins reported may reflect approximate responses that could correspond

to a wide range of possible distributions, including the two distributions shown. The wider

distribution could the correct one if the zero entries in the bins from 1 to 2 and from 4 to 5 are

8

merely the result of rounding down probabilities just under 5 percent, so that all reported

bins could be in multiples of 10 percent. But the narrower distribution could be correct if

no such rounding took place and uncertainty was indeed so small that the true probabilities

in the bins from 1 to 2 and from 4 to 5 are virtually zero. A minimization procedure that

matches the normal CDF to the empirical CDF, of course, cannot not recognize the first

possibility and its implied rounding errors. Instead, for any probabilistic forecast where all

the density is concentrated in two consecutive bins, it mechanically identifies the best fitting

normal distribution to be the one with infinitesimal (zero) variance and a mean located at

the point dividing the two consecutive bins. That is, if the approximate nature of the

probabilistic responses, and associated rounding of answers results in a 2-bin distribution,

fitting a normal density will result in a severe downward bias in the reported individual

uncertainty. Using the mean of individual standard deviations computed in this manner

would subsequently result in a significant bias in the estimate of average uncertainty.

One way to mitigate this problem is to model directly the distribution of the individual

uncertainties and treat the uncertainties of forecasters that assign a positive probability

only to one or two intervals as small but unobserved. Specifically, let us suppose that the

individual uncertainty originates from a Gamma distribution:

vi,h,t ∼ Gamma(αh,t; βh,t),

then the values of the mean of inflation uncertainty and disagreement about this uncertainty

are recovered from the parameters αh,t and βh,t. To the extent this distribution could be

fit by treating the variances of responses with one or two bins as small but unobservable,

the distortion to the estimation of the distribution could be avoided. This can be done

by fitting a truncated distribution to the individual variances over the range of variances

that are observable. We implement this by fitting a distribution to the empirical CDF of

the individual variances that exceed some threshold, recognizing that the CDF has positive

mass at the threshold, but treating variances below that threshold as unobservable. An

example is presented in Figure 7. More precisely, define C the chosen cutoff, then vi,h,t(cτ )

for all cτ > C are all the individual variances above the threshold and F (vi,h,t(c1)), ...

F (vi,h,t(cNc)), where c1, ...cNc are the right endpoints of the intervals in which the range of

uncertainty values has been discretized, are the empirical CDF of vi,h,t(cτ ) defined at these

9

endpoints. We then minimize over the distribution parameters θ1 and θ2:

minθ1,θ2

Nc∑

τ=1

[CDF (vi,h,t(cτ ); θ1 , θ2)− F (vi,h,t(cτ ))]2 ,

where CDF (vi,h,t(cτ ); θ1 , θ2) is the cumulative distribution function of the candidate model

chosen to fit the distribution of the individual uncertainties. Since the uncertainty can only

take positive values, the choice of candidate models is limited to those distributions that are

defined only on a positive support. Among possible candidates we examined the Gamma,

Chi-square and Lognormal but in the following we concentrate our attention on the Gamma,

which provided the best fit among these. The resulting parameters of the distributions we

estimate by quarter allow us to track the characteristics of the distribution of uncertainty

over time. Since we are interested in the average of σi,h,t = √vi,h,t and we model vi,h,t to

be Gamma distributed, we apply a change in variable to recover the mean and standard

deviation of σi,h,t. Recall, that if vGh,t = EG

h,t(vi,h,t) = αh,t · βh,t and V ar(vi,h,t) = αh,t · β2h,t,

where EG indicates the expected value under the Gamma distribution, then:

σGh,t = EG

h,t(σi,h,t) = β1/2h,t ·

Γ(αh,t + 1/2)Γ(αh,t)

,

φGh,t =

√EG

h,t

(σi,h,t − σG

h,t

)2=

√αh,t · βh,t −

(σG

h,t

)2

It follows that fitting directly the distribution of the individual uncertainties provides a

way to summarize the quarterly uncertainty that improves upon taking a simple average of

individual standard deviations by circumventing the problems associated with respondents

with just one or two bins.

We apply this methodology to all the individual measures of uncertainty previously

obtained, that is we assume:

vMi,h,t ∼ Gamma(αM

h,t; βMh,t); vU

i,h,t ∼ Gamma(αUh,t;β

Uh,t) and vN

i,h,t ∼ Gamma(αNh,t;β

Nh,t),

and derive the following measures:

σGMh,t = EG

h,t(σMi,h,t) =

(βM

h,t

)1/2 · Γ(αMh,t + 1/2)

Γ(αMh,t)

,

φGMh,t =

√EG

h,t

(σM

i,h,t − σGMh,t

)2=

√αM

h,t · βMh,t −

(σGM

h,t

)2,

10

σGUh,t = EG

h,t(σUi,h,t) =

(βU

h,t

)1/2 · Γ(αUh,t + 1/2)

Γ(αUh,t)

,

φGUh,t =

√EG

h,t

(σU

i,h,t − σGUh,t

)2=

√αU

h,t · βUh,t −

(σGU

h,t

)2,

σGNh,t = EG

h,t(σNi,h,t) =

(βN

h,t

)1/2 · Γ(αNh,t + 1/2)

Γ(αNh,t)

,

and

φGNh,t =

√EG

h,t

(σN

i,h,t − σGNh,t

)2=

√αN

h,t · βNh,t −

(σGN

h,t

)2.

where the double superscript indicates that the Gamma (G) methodology has been applied

to the individual uncertainties derived under the midpoint (M), uniform (U) and normal

(N) assumptions.

5 Characteristics of the cross-sectional distributions over time

Figures 8–14 summarize the results of the quarter-by-quarter estimation in our sample. Each

figure presents a seasonal time series of alternative estimates of µ, s, σ and φ corresponding

to different estimates of the individual means and variances. The time series are seasonal

in that the forecast horizons vary by quarter according to our “approximate” year-ahead

convention. Thus, in Q1, the forecast horizon is 4 quarters, in Q2 it is 3 quarters, in Q3 it

is 6 quarters and in Q4 it is 5 quarters.

The mean for the individual means as well as the dispersion of the individual means

do not appear to be sensitive the the estimation assumption. The midpoint and uniform

methods are, of course, identical. But in addition, as Figures 8 and 9 confirm, the normal

density method provides essentially similar results.

The three methods do result in somewhat different estimates of the characteristics of

the distributions of uncertainty, though the differences are small. Figure 10 compares the

mean uncertainty obtained by fitting the truncated Gamma distribution by quarter to

each of the three alternatives. Similarly, Figure 11 compares the disagreement regarding

individual uncertainty. The mean estimates for uncertainty are generally higher for the

11

uniform method, as would be expected. But the midpoint and normal methods do not have

a material difference in the mean.

Table 1 shows the correlations of the mean, uncertainty and disagreement measures for

the midpoint-Gamma estimates. As can be seen, they are all positive but those between

µ and s, and between σ and φ are the highest. Comparing the top and bottom panels

indicates that the estimates are essentially similar regardless of whether the aggregate in-

flation forecast attributes are based on only regular survey respondents or all respondents,

including irregular ones.

To compare the evolution of uncertainty and disagreement over time Figures 12-14

collect the three measures corresponding to each of the three methods we used to estimate

individual variances. The comovements of the three variables are essentially similar in each

figure so we can focus attention on just one. As can be seen, disagreement and uncertainty

co-move, but their evolution differs over time. To see more clearly the correlation of mean

uncertainty with the mean outlook ad the two measures of disagreement, Figures 15-17

show scatterplots of these variables. As can be seen, the correlation is evident regardless of

whether one concentrates attention on just one quarter each year (to circumvent seasonality

concerns) or all quarters together.

6 Inflation and Term Premia

To illustrate the potential usefulness of distinguishing the various attributes of the inflation

outlook, in this section we examine their relationship to term premia. For this analysis,

we utilize estimates of the forward term premium at the 2-year, 5-year and 10-year ahead

horizons obtained from an arbitrage-free three-factor model of the term structure (see Kim

and Orphanides, 2005). The quarterly time-series for each premium selects the premium

prevailing at the last business day of the middle month of the quarter, so the premia are

always measured after the survey for that quarter has been conducted.

Table 2 presents the simple correlations of these three premia with the pooled (seasonal)

measures of disagreement and uncertainty corresponding to the midpoint-Gamma estima-

tion. (Results from the other two methods are essentially similar and are thus not reported.)

As can be seen, the term premia at all three horizons shown are positively correlated with

12

the mean forecast for inflation as well as with mean uncertainty, and the disagreement about

both the mean and the uncertainty. However, the correlation in this sample is strongest

with the uncertainty.

To gauge the statistical significance of these correlations, and account for the seasonal

nature of the inflation forecast variables, Table 3 presents linear regressions of the 2-year

premium on each one of the various measures, allowing for quarter-specific intercepts and

slope coefficients. The row noted with p denotes the p-values associated with the hypothesis

that all slope coefficients equal zero, (based on a HAC-robust covariance). As can be

seen, each of the four attributes of the inflation forecast individually significantly correlates

with the 2-year premium, but the relationship is strongest for the mean uncertainty of the

forecast. As before, the estimates are essentially similar regardless of whether the aggregate

inflation forecast attributes are based on only regular survey respondents or all respondents,

including irregular ones.

13

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14

Table 1Correlations of Attributes of Distributions of Inflation Forecasts

(Midpoint-Gamma, pooled quarters)

Premium µ σ s φ

All respondentsµ 1.00σ 0.47 1.00s 0.66 0.39 1.00φ 0.25 0.65 0.25 1.00

At least 12 surveysµ 1.00σ 0.45 1.00s 0.67 0.37 1.00φ 0.24 0.63 0.23 1.00

Notes: µ and s are the mean and standard deviation of of the distribution of the individualmean forecasts. σ and φ are the mean and standard deviation of the distribution of theindividual standard deviations. The entries report the correlations of the quarterly time-series obtained by pooling together the attributes of the 4-, 3-, 6-, and 5-quarter-aheadforecasts available in the first, second, third and fourth quarter, respectively.

15

Table 2Simple Correlations with Term Premia

(Midpoint-Gamma, pooled quarters)

Premium µ σ s φ

All respondents2-year 0.40 0.59 0.25 0.345-year 0.51 0.66 0.34 0.3810-year 0.59 0.68 0.42 0.40

At least 12 surveys2-year 0.40 0.59 0.26 0.345-year 0.51 0.66 0.35 0.3710-year 0.59 0.69 0.40 0.41

Notes: See notes to Table 1.

16

Table 3Explaining Term Premia

(2-year premium,Midpoint-Gamma )

Q µ σ s φ(1) (2) (3) (4)

All respondents1 0.16 3.23 1.34 1.632 0.17 4.65 1.07 2.323 0.39 3.93 4.24 3.564 0.22 3.64 −0.31 3.05

R2 0.16 0.38 0.13 0.09p 0.041 0.001 0.008 0.005

At least 12 surveys1 0.17 3.21 1.34 1.442 0.17 4.49 1.18 2.373 0.39 3.78 3.40 3.214 0.22 3.79 −0.07 4.40

R2 0.16 0.39 0.11 0.10p 0.041 0.001 0.028 0.011

Notes: Each column in each panel reports the slope parameters, γX,Q of the regression:

τt =∑

Q=1,4

(γ0 + γX,QXt) ·Qt + ηt

where τ is the two-year forward term premium, and X is the attribute of the inflationforecast shown at the top of the column. The quarter-specific constants are not reported.p denotes the p-value of the hypothesis that all the slope parameters are zero.

17

Figure 1

Number of Survey Respondents

0

20

40

60

80

100

120

1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

All respondentsRegular (4+ surveys)Regular (8+ surveys)Regular (12+ surveys)

18

Figure 2

Two Examples of Individual Density Forecasts

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6 7 8 9

0.0

0.2

0.4

0.6

0.8

1.0

-1 0 1 2 3 4 5 6 7 8 9

19

Figure 3

Number of Non-Zero Bins by Forecast Horizon

Horizon

0

20

40

60

80

100

1 2 3 4 5 6 7 8

1 2 3 4 5+

20

Figure 4

Rounding by Forecast Horizon

Horizon

0

20

40

60

80

100

1 2 3 4 5 6 7 8

1% 5% 10%

21

Figure 5

Rounding by Number of Non-Zero Bins

Bins

0

20

40

60

80

100

1 2 3 4 5+

1% 5% 10%

22

Figure 6

Fitting a Normal Density: Two Examples

0.0

0.5

1.0

1.5

2.0

-1 0 1 2 3 4 5 6 7 8 9

0.0

0.5

1.0

1.5

2.0

-1 0 1 2 3 4 5 6 7 8 9

23

Figure 7

Gamma Distribution of Individual Variances

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00

24

Figure 8

Inflation Outlook

0

2

4

6

8

10

1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

Percent

NormalMidpoint/Uniform

25

Figure 9

Disagreement

0.0

0.5

1.0

1.5

2.0

1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

Percent

NormalMidpoint/Uniform

26

Figure 10

Uncertainty

0.0

0.5

1.0

1.5

2.0

1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

Percent

NormalMidpointUniform

27

Figure 11

Disagreement about uncertainty

0.0

0.5

1.0

1.5

2.0

1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

Percent

NormalMidpointUniform

28

Figure 12

Uncertainty and Disagreement(Normal-Gamma)

0.0

0.5

1.0

1.5

2.0

1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

Percent

UncertaintyDisagreement about meanDisagreement about uncertainty

29

Figure 13

Uncertainty and Disagreement(Midpoint-Gamma)

0.0

0.5

1.0

1.5

2.0

1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

Percent

UncertaintyDisagreement about meanDisagreement about uncertainty

30

Figure 14

Uncertainty and Disagreement(Uniform-Gamma)

0.0

0.5

1.0

1.5

2.0

1969 1973 1977 1981 1985 1989 1993 1997 2001 2005

Percent

UncertaintyDisagreement about meanDisagreement about uncertainty

31

Figure 15

Inflation Outlook and Uncertainty

0

2

4

6

8

10

0.4 0.6 0.8 1.0 1.2 1.4

Q1

Q2

Q3

Q4

Uncertainty

32

Figure 16

Disagreement about Outlook and Uncertainty

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.4 0.6 0.8 1.0 1.2 1.4

Q1

Q2

Q3

Q4

Uncertainty

33

Figure 17

Disagreement about Uncertainty and Uncertainty

0.0

0.2

0.4

0.6

0.8

1.0

0.4 0.6 0.8 1.0 1.2 1.4

Q1

Q2

Q3

Q4

Uncertainty

34

Figure 18

Term Premium

0

2

4

6

8

1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

Percent

Two yearsFive yearsTen years

35